TPTP Problem File: ITP039^2.p
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%------------------------------------------------------------------------------
% File : ITP039^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_472__7214396_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Coincidence/prob_472__7214396_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 352 ( 97 unt; 70 typ; 0 def)
% Number of atoms : 1086 ( 299 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 6949 ( 122 ~; 0 |; 353 &;6112 @)
% ( 0 <=>; 362 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 15 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 341 ( 341 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 62 usr; 3 con; 0-13 aty)
% Number of variables : 2353 ( 138 ^;2094 !; 7 ?;2353 :)
% ( 114 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:48.593
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
frechet_good_interp: $tType > $tType > $tType > $tType ).
thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
frechet_strm: $tType > $tType > $tType ).
thf(ty_t_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Syntax_Oformula,type,
formula: $tType > $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Syntax_Otrm,type,
trm: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_sz,type,
sz: $tType ).
thf(ty_tf_sf,type,
sf: $tType ).
thf(ty_tf_sc,type,
sc: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (52)
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
euclid1678468529ng_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
euclid1155270486miring:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
coinci1993344360de_fml:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
thf(sy_c_Denotational__Semantics_OIagree,type,
denotational_Iagree:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_OVagree,type,
denotational_Vagree:
!>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_Ofml__sem,type,
denotational_fml_sem:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( formula @ A @ B @ C ) > ( set @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ).
thf(sy_c_Denotational__Semantics_Oids_Ovalid,type,
denotational_valid:
!>[Sf: $tType,Sc: $tType,Sz: $tType] : ( ( formula @ Sf @ Sc @ Sz ) > $o ) ).
thf(sy_c_Denotational__Semantics_Ois__interp,type,
denota2077489681interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
freche457001096interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
frechet_cr_strm:
!>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
freche227871258interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
freche229654227interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
frechet_raw_term:
!>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
frechet_simple_term:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_Ids_Oids,type,
ids:
!>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Static__Semantics_OFVF,type,
static_FVF:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OSIGF,type,
static_SIGF:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
thf(sy_c_Syntax_OEquiv,type,
equiv:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_OImplies,type,
implies:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_OPredicational,type,
predicational:
!>[B: $tType,A: $tType,C: $tType] : ( B > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_Odfree,type,
dfree:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
thf(sy_c_Syntax_Oids_OP,type,
p:
!>[Sc: $tType,Sf: $tType,Sz: $tType] : ( Sc > ( formula @ Sf @ Sc @ Sz ) ) ).
thf(sy_c_Syntax_Oids_Osingleton,type,
singleton:
!>[Sz: $tType,A: $tType] : ( Sz > ( trm @ A @ Sz ) > Sz > ( trm @ A @ Sz ) ) ).
thf(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__unique,type,
right_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v__092_060phi_062,type,
phi: formula @ a @ b @ c ).
% Relevant facts (256)
thf(fact_0_coincide__fml__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( coinci1993344360de_fml @ A @ B @ C )
= ( ^ [Phi: formula @ A @ B @ C] :
! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( denotational_Iagree @ A @ B @ C @ I @ J @ ( static_SIGF @ A @ B @ C @ Phi ) )
=> ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVF @ A @ B @ C @ Phi ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu @ ( denotational_fml_sem @ A @ B @ C @ I @ Phi ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu2 @ ( denotational_fml_sem @ A @ B @ C @ J @ Phi ) ) ) ) ) ) ) ) ).
% coincide_fml_def
thf(fact_1_ids_Ocoincide__fml__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( coinci1993344360de_fml @ A @ B @ C @ Phi2 )
= ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( denotational_Iagree @ A @ B @ C @ I @ J @ ( static_SIGF @ A @ B @ C @ Phi2 ) )
=> ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVF @ A @ B @ C @ Phi2 ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu @ ( denotational_fml_sem @ A @ B @ C @ I @ Phi2 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu2 @ ( denotational_fml_sem @ A @ B @ C @ J @ Phi2 ) ) ) ) ) ) ) ) ) ).
% ids.coincide_fml_def
thf(fact_2_Iagree__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A2: denota1663640101rp_ext @ A @ B @ C @ product_unit,B2: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
( ( denotational_Iagree @ A @ B @ C @ A2 @ B2 @ V )
=> ( denotational_Iagree @ A @ B @ C @ B2 @ A2 @ V ) ) ) ).
% Iagree_comm
thf(fact_3_Iagree__refl,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,A2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I2 @ I2 @ A2 ) ) ).
% Iagree_refl
thf(fact_4_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A2 @ B2 @ V )
=> ( denotational_Vagree @ A @ B2 @ A2 @ V ) ) ) ).
% agree_comm
thf(fact_5_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A2: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu3 @ Nu3 @ A2 ) ) ).
% agree_refl
thf(fact_6_iff__to__impl,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) )
= ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) )
& ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) ) ) ) ) ) ).
% iff_to_impl
thf(fact_7_iff__sem,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( equiv @ B @ C @ A @ A2 @ B2 ) ) )
= ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) ) ) ) ).
% iff_sem
thf(fact_8_raw__interp__inject,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
( ( ( freche229654227interp @ A @ B @ C @ X )
= ( freche229654227interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ).
% raw_interp_inject
thf(fact_9_raw__term__inject,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
( ( ( frechet_raw_term @ A @ C @ X )
= ( frechet_raw_term @ A @ C @ Y ) )
= ( X = Y ) ) ).
% raw_term_inject
thf(fact_10_impl__sem,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( implies @ B @ C @ A @ A2 @ B2 ) ) )
= ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) ) ) ) ).
% impl_sem
thf(fact_11_P__def,axiom,
( ( p @ sc @ sf @ sz )
= ( predicational @ sc @ sf @ sz ) ) ).
% P_def
thf(fact_12_seq__sem_Ocases,axiom,
! [X: product_prod @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) )] :
~ ! [I3: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,S: product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) )] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) ) @ I3 @ S ) ) ).
% seq_sem.cases
thf(fact_13_ids_Oseq__sem_Ocases,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I3 @ S ) ) ) ) ).
% ids.seq_sem.cases
thf(fact_14_ids_Oseq__sem_Oinduct,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I3 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ) ).
% ids.seq_sem.induct
thf(fact_15_FunctionFrechet_Ocases,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [X: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
~ ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,I4: A] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I3 @ I4 ) ) ) ).
% FunctionFrechet.cases
thf(fact_16_ids_OP__def,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p @ Sc @ Sf @ Sz @ P2 )
= ( predicational @ Sc @ Sf @ Sz @ P2 ) ) ) ) ).
% ids.P_def
thf(fact_17_raw__interp__inverse,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C] :
( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
= X ) ) ).
% raw_interp_inverse
thf(fact_18_raw__term__inverse,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
= X ) ).
% raw_term_inverse
thf(fact_19_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_21_ids_Oraw__term__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( frechet_raw_term @ A @ C @ X )
= ( frechet_raw_term @ A @ C @ Y ) )
= ( X = Y ) ) ) ) ).
% ids.raw_term_inject
thf(fact_22_ids_Oraw__interp__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( freche229654227interp @ A @ B @ C @ X )
= ( freche229654227interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ) ).
% ids.raw_interp_inject
thf(fact_23_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_24_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_25_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_26_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_27_ids_Osingleton_Ocases,axiom,
! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( trm @ A @ Sz ) @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [T2: trm @ A @ Sz,I4: Sz] :
( X
!= ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T2 @ I4 ) ) ) ) ).
% ids.singleton.cases
thf(fact_28_ids_Oraw__interp__inverse,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
= X ) ) ) ).
% ids.raw_interp_inverse
thf(fact_29_ids_Oraw__term__inverse,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
= X ) ) ) ).
% ids.raw_term_inverse
thf(fact_30_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_31_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_32_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_33_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_34_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_35_ids_Osingleton_Oinduct,axiom,
! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [T2: trm @ A @ Sz,X_1: Sz] : ( P @ T2 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ) ).
% ids.singleton.induct
thf(fact_36_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_37_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_38_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_39_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_40_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_41_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_42_euclid__ext__aux_Ocases,axiom,
! [A: $tType] :
( ( euclid1678468529ng_gcd @ A )
=> ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
~ ! [S2: A,S3: A,T3: A,T2: A,R: A,R2: A] :
( X
!= ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T2 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% euclid_ext_aux.cases
thf(fact_43_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_44_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C3 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G3 @ X3 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_ids_Ovne23,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ) ).
% ids.vne23
thf(fact_50_ids_Ovne13,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid3 ) ) ) ).
% ids.vne13
thf(fact_51_ids_Ovne12,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid2 ) ) ) ).
% ids.vne12
thf(fact_52_ids_Opne34,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid4 ) ) ) ).
% ids.pne34
thf(fact_53_ids_Opne24,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid4 ) ) ) ).
% ids.pne24
thf(fact_54_ids_Opne23,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid3 ) ) ) ).
% ids.pne23
thf(fact_55_gcd_Ocases,axiom,
! [A: $tType] :
( ( euclid1155270486miring @ A )
=> ! [X: product_prod @ A @ A] :
~ ! [A5: A,B5: A] :
( X
!= ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% gcd.cases
thf(fact_56_ids_Oid__simps_I24_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid3 ) ) ) ).
% ids.id_simps(24)
thf(fact_57_ids_Oid__simps_I23_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid2 ) ) ) ).
% ids.id_simps(23)
thf(fact_58_ids_Oid__simps_I22_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid1 ) ) ) ).
% ids.id_simps(22)
thf(fact_59_ids_Oid__simps_I21_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid1 ) ) ) ).
% ids.id_simps(21)
thf(fact_60_ids_Oid__simps_I20_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid2 ) ) ) ).
% ids.id_simps(20)
thf(fact_61_ids_Oid__simps_I19_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid1 ) ) ) ).
% ids.id_simps(19)
thf(fact_62_ids_Oid__simps_I18_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid3 != Fid1 ) ) ) ).
% ids.id_simps(18)
thf(fact_63_ids_Oid__simps_I17_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid3 != Fid2 ) ) ) ).
% ids.id_simps(17)
thf(fact_64_ids_Oid__simps_I16_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid1 ) ) ) ).
% ids.id_simps(16)
thf(fact_65_ids_Oid__simps_I15_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid3 != Vid1 ) ) ) ).
% ids.id_simps(15)
thf(fact_66_ids_Oid__simps_I14_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid3 != Vid2 ) ) ) ).
% ids.id_simps(14)
thf(fact_67_ids_Oid__simps_I13_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid1 ) ) ) ).
% ids.id_simps(13)
thf(fact_68_ids_Oid__simps_I12_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid4 ) ) ) ).
% ids.id_simps(12)
thf(fact_69_ids_Oid__simps_I11_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid4 ) ) ) ).
% ids.id_simps(11)
thf(fact_70_ids_Oid__simps_I10_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid4 ) ) ) ).
% ids.id_simps(10)
thf(fact_71_ids_Oid__simps_I9_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid3 ) ) ) ).
% ids.id_simps(9)
thf(fact_72_ids_Oid__simps_I8_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid3 ) ) ) ).
% ids.id_simps(8)
thf(fact_73_ids_Oid__simps_I7_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid2 ) ) ) ).
% ids.id_simps(7)
thf(fact_74_ids_Oid__simps_I6_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid3 ) ) ) ).
% ids.id_simps(6)
thf(fact_75_ids_Oid__simps_I5_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid3 ) ) ) ).
% ids.id_simps(5)
thf(fact_76_ids_Oid__simps_I4_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid2 ) ) ) ).
% ids.id_simps(4)
thf(fact_77_ids_Oid__simps_I3_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid3 ) ) ) ).
% ids.id_simps(3)
thf(fact_78_ids_Oid__simps_I2_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ) ).
% ids.id_simps(2)
thf(fact_79_ids_Oid__simps_I1_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid2 ) ) ) ).
% ids.id_simps(1)
thf(fact_80_ids__def,axiom,
! [Sf: $tType,Sc: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ( ( ids @ Sz @ Sf @ Sc )
= ( ^ [Vid12: Sz,Vid22: Sz,Vid32: Sz,Fid12: Sf,Fid22: Sf,Fid32: Sf,Pid12: Sc,Pid22: Sc,Pid32: Sc,Pid42: Sc] :
( ( Vid12 != Vid22 )
& ( Vid22 != Vid32 )
& ( Vid12 != Vid32 )
& ( Fid12 != Fid22 )
& ( Fid22 != Fid32 )
& ( Fid12 != Fid32 )
& ( Pid12 != Pid22 )
& ( Pid22 != Pid32 )
& ( Pid12 != Pid32 )
& ( Pid12 != Pid42 )
& ( Pid22 != Pid42 )
& ( Pid32 != Pid42 ) ) ) ) ) ).
% ids_def
thf(fact_81_ids_Ofne12,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid2 ) ) ) ).
% ids.fne12
thf(fact_82_ids_Ofne13,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid3 ) ) ) ).
% ids.fne13
thf(fact_83_ids_Ofne23,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid3 ) ) ) ).
% ids.fne23
thf(fact_84_ids_Ointro,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( Vid1 != Vid2 )
=> ( ( Vid2 != Vid3 )
=> ( ( Vid1 != Vid3 )
=> ( ( Fid1 != Fid2 )
=> ( ( Fid2 != Fid3 )
=> ( ( Fid1 != Fid3 )
=> ( ( Pid1 != Pid2 )
=> ( ( Pid2 != Pid3 )
=> ( ( Pid1 != Pid3 )
=> ( ( Pid1 != Pid4 )
=> ( ( Pid2 != Pid4 )
=> ( ( Pid3 != Pid4 )
=> ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ids.intro
thf(fact_85_ids_Opne12,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid2 ) ) ) ).
% ids.pne12
thf(fact_86_ids_Opne13,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid3 ) ) ) ).
% ids.pne13
thf(fact_87_ids_Opne14,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid4 ) ) ) ).
% ids.pne14
thf(fact_88_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A6: B,B6: C] : ( F4 @ ( product_Pair @ B @ C @ A6 @ B6 ) ) ) ) ).
% curry_conv
thf(fact_89_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S4: B,R4: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S4 ) @ R4 )
=> ( ( S5 = S4 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_90_curryI,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 ) ) ).
% curryI
thf(fact_91_ids_Oproj__sing1,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( singleton @ Sz @ A @ Vid1 @ Theta @ Vid1 )
= Theta ) ) ) ).
% ids.proj_sing1
thf(fact_92_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_93_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_94_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_95_ids_Ovalid__def,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ Sf @ Sc @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denotational_valid @ Sf @ Sc @ Sz @ Phi2 )
= ( ! [I: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,Nu: product_prod @ ( finite_Cartesian_vec @ real @ Sz ) @ ( finite_Cartesian_vec @ real @ Sz )] :
( ( denota2077489681interp @ Sf @ Sc @ Sz @ I )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ Sz ) @ ( finite_Cartesian_vec @ real @ Sz ) ) @ Nu @ ( denotational_fml_sem @ Sf @ Sc @ Sz @ I @ Phi2 ) ) ) ) ) ) ) ).
% ids.valid_def
thf(fact_96_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_97_ids_Osingleton_Ocong,axiom,
! [A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ( ( singleton @ Sz @ A )
= ( singleton @ Sz @ A ) ) ) ).
% ids.singleton.cong
thf(fact_98_curryE,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 )
=> ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryE
thf(fact_99_curryD,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 )
=> ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryD
thf(fact_100_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_101_good__interp__inverse,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
= Y ) ) ) ).
% good_interp_inverse
thf(fact_102_ids_Ogood__interp__inverse,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
= Y ) ) ) ) ).
% ids.good_interp_inverse
thf(fact_103_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_104_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_105_raw__interp__induct,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% raw_interp_induct
thf(fact_106_raw__interp__cases,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ~ ! [X3: frechet_good_interp @ A @ B @ C] :
( Y
!= ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ).
% raw_interp_cases
thf(fact_107_raw__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C] : ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% raw_interp
thf(fact_108_good__interp__inject,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( ( freche227871258interp @ A @ B @ C @ X )
= ( freche227871258interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ) ) ).
% good_interp_inject
thf(fact_109_good__interp__induct,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% good_interp_induct
thf(fact_110_good__interp__cases,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C] :
~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( X
= ( freche227871258interp @ A @ B @ C @ Y3 ) )
=> ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% good_interp_cases
thf(fact_111_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( member @ C @ Z @ ( C3 @ A5 @ B5 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_112_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z @ ( C3 @ A3 @ B3 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_113_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
( ! [A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B5 )
= P2 )
=> ( C3 @ A5 @ B5 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X ) ) ).
% case_prodI2'
thf(fact_114_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( C3 @ A5 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_115_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
( ( F3 @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_116_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F3 @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_117_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
= F3 ) ).
% curry_case_prod
thf(fact_118_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
= F3 ) ).
% case_prod_curry
thf(fact_119_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P2: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y4: B,X4: C] : ( F3 @ X4 @ Y4 )
@ ( product_swap @ C @ B @ P2 ) )
= ( product_case_prod @ C @ B @ A @ F3 @ P2 ) ) ).
% case_swap
thf(fact_120_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A2: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A2 )
=> ( member @ C @ ( F3 @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A2 ) ) ) ).
% pair_imageI
thf(fact_121_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_122_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C3: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C3 )
= ( ^ [X4: A,Y4: B] : C3 ) ) ).
% curry_K
thf(fact_123_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F3 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_124_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_125_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X22: B] : ( H @ ( F3 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_126_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F3 @ X3 @ Y3 )
= ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_127_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_128_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_129_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_130_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A3: A,B3: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C3 )
=> ( R4 @ A3 @ B3 @ C3 ) ) ).
% case_prodD'
thf(fact_131_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_132_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F3 @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_133_ids_Oraw__interp__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
=> ( P @ Y ) ) ) ) ) ).
% ids.raw_interp_induct
thf(fact_134_ids_Oraw__interp__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ~ ! [X3: frechet_good_interp @ A @ B @ C] :
( Y
!= ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ) ).
% ids.raw_interp_cases
thf(fact_135_ids_Oraw__interp,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% ids.raw_interp
thf(fact_136_ids_Ogood__interp__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( X
= ( freche227871258interp @ A @ B @ C @ Y3 ) )
=> ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ) ).
% ids.good_interp_cases
thf(fact_137_ids_Ogood__interp__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
=> ( P @ X ) ) ) ) ).
% ids.good_interp_induct
thf(fact_138_ids_Ogood__interp__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( ( freche227871258interp @ A @ B @ C @ X )
= ( freche227871258interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% ids.good_interp_inject
thf(fact_139_cr__strm__def,axiom,
! [B: $tType,A: $tType] :
( ( frechet_cr_strm @ A @ B )
= ( ^ [X4: trm @ A @ B,Y4: frechet_strm @ A @ B] :
( X4
= ( frechet_raw_term @ A @ B @ Y4 ) ) ) ) ).
% cr_strm_def
thf(fact_140_cr__good__interp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( freche457001096interp @ A @ B @ C )
= ( ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y4: frechet_good_interp @ A @ B @ C] :
( X4
= ( freche229654227interp @ A @ B @ C @ Y4 ) ) ) ) ) ).
% cr_good_interp_def
thf(fact_141_image__ident,axiom,
! [A: $tType,Y5: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_142_type__definition__good__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% type_definition_good_interp
thf(fact_143_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X: B,A2: set @ B] :
( ( B3
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_144_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B6: B] :
( P
& ( Q @ A6 @ B6 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_145_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_146_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F3: A > B] :
( ( member @ A @ X @ A2 )
=> ( ( B3
= ( F3 @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F3 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_147_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A2 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_148_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F3 @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_149_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_150_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F3: B > A,A2: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A2 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A2 )
& ( Z
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_151_imageI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,F3: A > B] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A2 ) ) ) ).
% imageI
thf(fact_152_ids_Ocr__strm__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( frechet_cr_strm @ A @ B )
= ( ^ [X4: trm @ A @ B,Y4: frechet_strm @ A @ B] :
( X4
= ( frechet_raw_term @ A @ B @ Y4 ) ) ) ) ) ) ).
% ids.cr_strm_def
thf(fact_153_ids_Ocr__good__interp__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( freche457001096interp @ A @ B @ C )
= ( ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y4: frechet_good_interp @ A @ B @ C] :
( X4
= ( freche229654227interp @ A @ B @ C @ Y4 ) ) ) ) ) ) ).
% ids.cr_good_interp_def
thf(fact_154_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A2 ) )
& ( P @ X4 ) ) )
= ( image @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A2 )
& ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_155_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A2: set @ C] :
( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A2 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_156_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F3 @ X3 ) )
=> ~ ( member @ B @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_157_ids_Otype__definition__good__interp,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% ids.type_definition_good_interp
thf(fact_158_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R5: C] : ( F4 @ L @ R5 @ Y4 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_159_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F3 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P2 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_160_simple__term__inverse,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_161_strm_Orep__transfer,axiom,
! [C: $tType,A: $tType] :
( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
@ ^ [Y6: trm @ A @ C,Z2: trm @ A @ C] : Y6 = Z2
@ ^ [X4: trm @ A @ C] : X4
@ ( frechet_raw_term @ A @ C ) ) ).
% strm.rep_transfer
thf(fact_162_raw__term__induct,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_term_induct
thf(fact_163_raw__term__cases,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ~ ! [X3: frechet_strm @ A @ C] :
( Y
!= ( frechet_raw_term @ A @ C @ X3 ) ) ) ).
% raw_term_cases
thf(fact_164_raw__term,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C] : ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% raw_term
thf(fact_165_simple__term__inject,axiom,
! [C: $tType,A: $tType,X: trm @ A @ C,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( ( frechet_simple_term @ A @ C @ X )
= ( frechet_simple_term @ A @ C @ Y ) )
= ( X = Y ) ) ) ) ).
% simple_term_inject
thf(fact_166_simple__term__induct,axiom,
! [C: $tType,A: $tType,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
( ! [Y3: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
=> ( P @ X ) ) ).
% simple_term_induct
thf(fact_167_simple__term__cases,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
~ ! [Y3: trm @ A @ C] :
( ( X
= ( frechet_simple_term @ A @ C @ Y3 ) )
=> ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ).
% simple_term_cases
thf(fact_168_ids_Oraw__term,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% ids.raw_term
thf(fact_169_ids_Oraw__term__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ~ ! [X3: frechet_strm @ A @ C] :
( Y
!= ( frechet_raw_term @ A @ C @ X3 ) ) ) ) ) ).
% ids.raw_term_cases
thf(fact_170_ids_Oraw__term__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
=> ( P @ Y ) ) ) ) ) ).
% ids.raw_term_induct
thf(fact_171_ids_Osimple__term__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [Y3: trm @ A @ C] :
( ( X
= ( frechet_simple_term @ A @ C @ Y3 ) )
=> ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ) ).
% ids.simple_term_cases
thf(fact_172_ids_Osimple__term__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [Y3: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
=> ( P @ X ) ) ) ) ).
% ids.simple_term_induct
thf(fact_173_ids_Osimple__term__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ C,Y: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( ( frechet_simple_term @ A @ C @ X )
= ( frechet_simple_term @ A @ C @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% ids.simple_term_inject
thf(fact_174_ids_Ostrm_Orep__transfer,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
@ ^ [Y6: trm @ A @ C,Z2: trm @ A @ C] : Y6 = Z2
@ ^ [X4: trm @ A @ C] : X4
@ ( frechet_raw_term @ A @ C ) ) ) ) ).
% ids.strm.rep_transfer
thf(fact_175_ids_Osimple__term__inverse,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
= Y ) ) ) ) ).
% ids.simple_term_inverse
thf(fact_176_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_177_type__definition__strm,axiom,
! [C: $tType,A: $tType] : ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% type_definition_strm
thf(fact_178_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( T4
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T4
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [X4: A] : X4
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_179_good__interp_Orep__transfer,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
@ ^ [Y6: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z2: denota1663640101rp_ext @ A @ B @ C @ product_unit] : Y6 = Z2
@ ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X4
@ ( freche229654227interp @ A @ B @ C ) ) ) ).
% good_interp.rep_transfer
thf(fact_180_ids_Ogood__interp_Orep__transfer,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
@ ^ [Y6: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z2: denota1663640101rp_ext @ A @ B @ C @ product_unit] : Y6 = Z2
@ ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X4
@ ( freche229654227interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.rep_transfer
thf(fact_181_ids_Otype__definition__strm,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% ids.type_definition_strm
thf(fact_182_good__interp_Odomain,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
= ( denota2077489681interp @ A @ B @ C ) ) ) ).
% good_interp.domain
thf(fact_183_strm_Odomain,axiom,
! [C: $tType,A: $tType] :
( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
= ( dfree @ A @ C ) ) ).
% strm.domain
thf(fact_184_case__prod__const,axiom,
! [C: $tType,B: $tType,A: $tType,C3: C] :
( ( product_case_prod @ A @ B @ C
@ ^ [I5: A,J2: B] : C3 )
= ( ^ [Uu: product_prod @ A @ B] : C3 ) ) ).
% case_prod_const
thf(fact_185_ids_Ostrm_Odomain,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
= ( dfree @ A @ C ) ) ) ) ).
% ids.strm.domain
thf(fact_186_ids_Ogood__interp_Odomain,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
= ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.domain
thf(fact_187_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_188_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_189_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S6 ) ) )
= ( R4 = S6 ) ) ).
% pred_equals_eq2
thf(fact_190_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C5 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C5 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_191_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C5 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C5 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_192_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A2: set @ C,B2: set @ D] :
( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
@ ( product_Sigma @ C @ D @ A2
@ ^ [Uu: C] : B2 ) )
= ( product_Sigma @ A @ B @ ( image @ C @ A @ F3 @ A2 )
@ ^ [Uu: A] : ( image @ D @ B @ G3 @ B2 ) ) ) ).
% image_paired_Times
thf(fact_193_strm_Oright__unique,axiom,
! [C: $tType,A: $tType] : ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% strm.right_unique
thf(fact_194_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
= ( ( member @ A @ A3 @ A2 )
& ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_195_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A3 @ A2 )
=> ( ( member @ B @ B3 @ ( B2 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_196_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B6: B] :
( ( P @ A6 )
& ( Q @ B6 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_197_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( P @ X4 )
& ( Q @ X4 @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_198_typedef__right__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( T4
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( right_unique @ A @ B @ T4 ) ) ) ).
% typedef_right_unique
thf(fact_199_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ~ ( ( member @ A @ A3 @ A2 )
=> ~ ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_200_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ).
% SigmaD2
thf(fact_201_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ( member @ A @ A3 @ A2 ) ) ).
% SigmaD1
thf(fact_202_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B2 @ X3 ) )
=> ( C3
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_203_Sigma__cong,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A2 = B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( C5 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A2 @ C5 )
= ( product_Sigma @ A @ B @ B2 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_204_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A2: set @ B,B2: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A2
@ ^ [Uu: B] : C5 )
= ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : C5 ) )
= ( A2 = B2 ) ) ) ).
% Times_eq_cancel2
thf(fact_205_ids_Ostrm_Oright__unique,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% ids.strm.right_unique
thf(fact_206_product__swap,axiom,
! [B: $tType,A: $tType,A2: set @ B,B2: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A2
@ ^ [Uu: B] : B2 ) )
= ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : A2 ) ) ).
% product_swap
thf(fact_207_swap__product,axiom,
! [B: $tType,A: $tType,A2: set @ B,B2: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I5: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I5 ) )
@ ( product_Sigma @ B @ A @ A2
@ ^ [Uu: B] : B2 ) )
= ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : A2 ) ) ).
% swap_product
thf(fact_208_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A7: set @ A,B7: set @ B] :
( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B7 ) ) ) ).
% Product_Type.product_def
thf(fact_209_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A2 @ B2 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A2
@ ^ [Uu: A] : B2 ) ) ) ).
% member_product
thf(fact_210_good__interp_Oright__unique,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% good_interp.right_unique
thf(fact_211_ids_Ogood__interp_Oright__unique,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.right_unique
thf(fact_212_strm_Oleft__unique,axiom,
! [C: $tType,A: $tType] : ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% strm.left_unique
thf(fact_213_good__interp_Oleft__unique,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% good_interp.left_unique
thf(fact_214_ids_Ogood__interp_Oleft__unique,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.left_unique
thf(fact_215_ids_Ostrm_Oleft__unique,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% ids.strm.left_unique
thf(fact_216_typedef__left__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( T4
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( left_unique @ A @ B @ T4 ) ) ) ).
% typedef_left_unique
thf(fact_217_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_218_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F3: B > A,A2: set @ B,A8: set @ A,G3: D > C,B2: set @ D,B8: set @ C] :
( ( ( image @ B @ A @ F3 @ A2 )
= A8 )
=> ( ( ( image @ D @ C @ G3 @ B2 )
= B8 )
=> ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
@ ( product_Sigma @ B @ D @ A2
@ ^ [Uu: B] : B2 ) )
= ( product_Sigma @ A @ C @ A8
@ ^ [Uu: A] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_219_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y4: B] : Y4 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_220_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A3: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
= ( product_Pair @ A @ B @ ( F3 @ A3 ) @ ( G3 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_221_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R3 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_222_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R4: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R4 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R4 ) ) ) ).
% map_prod_imageI
thf(fact_223_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R4: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R4 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R4 ) ) ) ).
% prod_fun_imageE
thf(fact_224_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F3: D > B,G3: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F3 @ G3 @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R5: E] : ( H @ ( F3 @ L ) @ ( G3 @ R5 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_225_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X4: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_226_map__prod__image,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A2: set @ C,B2: set @ D] :
( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 )
@ ( product_Sigma @ C @ D @ A2
@ ^ [Uu: C] : B2 ) )
= ( product_Sigma @ A @ B @ ( image @ C @ A @ F3 @ A2 )
@ ^ [Uu: A] : ( image @ D @ B @ G3 @ B2 ) ) ) ).
% map_prod_image
thf(fact_227_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S7 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_228_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4
@ T5 )
= T5 ) ).
% prod.map_ident
thf(fact_229_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_230_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_231_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F3 @ X3 ) ) ) ).
% rangeE
thf(fact_232_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_233_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G3: B > C] :
( ( image @ B @ A
@ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F3 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_234_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X: B] :
( ( B3
= ( F3 @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_235_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_236_fun_Orel__refl__strong,axiom,
! [A: $tType,B: $tType,X: B > A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image @ B @ A @ X @ ( top_top @ ( set @ B ) ) ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_rel_fun @ B @ B @ A @ A
@ ^ [Y6: B,Z2: B] : Y6 = Z2
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl_strong
thf(fact_237_fun_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,R4: A > B > $o,X: D > A,Y: D > B,Ra: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ R4
@ X
@ Y )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image @ D @ B @ Y @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R4 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ Ra
@ X
@ Y ) ) ) ).
% fun.rel_mono_strong
thf(fact_238_fun_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,X: D > A,Ya: D > A,Y: D > B,Xa: D > B,R4: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image @ D @ B @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R4 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ R4
@ X
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ Ra
@ Ya
@ Xa ) ) ) ) ) ).
% fun.rel_cong
thf(fact_239_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_240_type__definition_Ouniv,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( top_top @ ( set @ B ) )
= ( image @ A @ B @ Abs @ A2 ) ) ) ).
% type_definition.univ
thf(fact_241_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( image @ A @ B @ Abs @ A2 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_242_Vagree__univ,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: finite_Cartesian_vec @ real @ A,B3: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D4: finite_Cartesian_vec @ real @ A] :
( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A3 @ B3 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D4 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
=> ( ( A3 = C3 )
& ( B3 = D4 ) ) ) ) ).
% Vagree_univ
thf(fact_243_agree__UNIV__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( denotational_Vagree @ A @ Nu3 @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
=> ( Nu3 = Omega ) ) ) ).
% agree_UNIV_eq
thf(fact_244_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( image @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
= A2 ) ) ).
% type_definition.Rep_range
thf(fact_245_surjD,axiom,
! [A: $tType,B: $tType,F3: B > A,Y: A] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F3 @ X3 ) ) ) ).
% surjD
thf(fact_246_surjE,axiom,
! [A: $tType,B: $tType,F3: B > A,Y: A] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F3 @ X3 ) ) ) ).
% surjE
thf(fact_247_surj__def,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X4: B] :
( Y4
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_248_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F3: A > B] :
( ! [X3: A] :
( ( G3 @ ( F3 @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_249_top__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( top @ B )
& ( top @ A ) )
=> ( ( top_top @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% top_prod_def
thf(fact_250_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_251_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
= ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_252_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_253_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_254_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_255_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
% Subclasses (2)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
% Type constructors (16)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 )
=> ( top @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
top @ $o ).
thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
linorder @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_6,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_8,axiom,
! [A9: $tType,A10: $tType] :
( ( ( top @ A9 )
& ( top @ A10 ) )
=> ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_9,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_10,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_11,axiom,
top @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_12,axiom,
! [A9: $tType,A10: $tType] :
( ( ( linorder @ A9 )
& ( cARD_1 @ A10 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_13,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% Free types (7)
thf(tfree_0,hypothesis,
finite_finite @ sz ).
thf(tfree_1,hypothesis,
linorder @ sz ).
thf(tfree_2,hypothesis,
finite_finite @ sf ).
thf(tfree_3,hypothesis,
finite_finite @ sc ).
thf(tfree_4,hypothesis,
finite_finite @ b ).
thf(tfree_5,hypothesis,
finite_finite @ a ).
thf(tfree_6,hypothesis,
finite_finite @ c ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGF @ a @ b @ c @ phi ) )
=> ( ( denotational_Vagree @ c @ Nu @ Nu2 @ ( static_FVF @ a @ b @ c @ phi ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ ( denotational_fml_sem @ a @ b @ c @ I @ phi ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ ( denotational_fml_sem @ a @ b @ c @ J @ phi ) ) ) ) ) )
= ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGF @ a @ b @ c @ phi ) )
=> ( ( denotational_Vagree @ c @ Nu @ Nu2 @ ( static_FVF @ a @ b @ c @ phi ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ ( denotational_fml_sem @ a @ b @ c @ I @ phi ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ ( denotational_fml_sem @ a @ b @ c @ J @ phi ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------